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Department of Mathematics

Question

Math 6520 • Algebraic Topology

Fall 2020 • Question 2

Question

Math 6520 • Algebraic Topology Fall 2020 • Question 2

Let \(x_0\in \mathbb {R}P^2\) be a basepoint and view the wedge \(\mathbb {R}P^2\vee \mathbb {R}P^2\) as the subspace of the product \(\mathbb {R}P^2\times \mathbb {R}P^2\) where at least one coordinate is \(x_0\).

  1. Use the Seifert-van Kampen theorem to compute \(\pi _1(\mathbb {R}P^2\vee \mathbb {R}P^2)\).
  2. Prove that \(\mathbb {R}P^2\vee \mathbb {R}P^2\) is not a retract of \(\mathbb {R}P^2\times \mathbb {R}P^2\).

Raw LaTeX

Let \(x_0\in\mathbb{R}P^2\) be a basepoint and view the wedge \(\mathbb{R}P^2\vee\mathbb{R}P^2\) as the
subspace of the product \(\mathbb{R}P^2\times\mathbb{R}P^2\) where at least one coordinate is \(x_0\).
\begin{enumerate}

\item
Use the Seifert-van Kampen theorem to compute \(\pi_1(\mathbb{R}P^2\vee\mathbb{R}P^2)\).


\item
Prove that \(\mathbb{R}P^2\vee\mathbb{R}P^2\) is not a retract of \(\mathbb{R}P^2\times\mathbb{R}P^2\).

\end{enumerate}

Topic Tags

Fundamental Groups Homotopy Equivalence, Retracts, and Contractibility Homotopy Groups Seifert-van Kampen Theorem